Enhancing QoS Through Fluid Antenna Systems over Correlated Nakagami-m Fading Channels
Item Type Conference Paper
Authors Tlebaldiyeva, Leila;Nauryzbayev, Galymzhan;Arzykulov, Sultangali;Eltawil, Ahmed;Tsiftsis, Theodoros
Citation Tlebaldiyeva, L., Nauryzbayev, G., Arzykulov, S., Eltawil, A., &
Tsiftsis, T. (2022). Enhancing QoS Through Fluid Antenna Systems over Correlated Nakagami-m Fading Channels. 2022 IEEE
Wireless Communications and Networking Conference (WCNC).
https://doi.org/10.1109/wcnc51071.2022.9771633 Eprint version Post-print
DOI 10.1109/wcnc51071.2022.9771633
Publisher IEEE
Rights Archived with thanks to IEEE Download date 2023-12-23 17:50:43
Link to Item http://hdl.handle.net/10754/678034
Enhancing QoS Through Fluid Antenna Systems over Correlated Nakagami- m Fading Channels
Leila Tlebaldiyeva, Galymzhan Nauryzbayev, ◦Sultangali Arzykulov, ◦Ahmed Eltawil, and§#Theodoros Tsiftsis School of Engineering and Digital Sciences, Nazarbayev University, Nur-Sultan, Z05H0K3, Kazakhstan
◦CEMSE Division, King Abdullah University of Science and Technology, Thuwal, KSA 23955-6900
§Dept. of Computer Science & Telecommunications, University of Thessaly, Greece
#School of Intelligent Systems Science and Engineering, Jinan University, China
Emails:{ltlebaldiyeva, galymzhan.nauryzbayev}@nu.edu.kz,◦{sultangali.arzykulov, ahmed.eltawil}@kaust.edu.sa,
Abstract—Fluid antenna systems (FAS) enable mechanically flexible antennas that offer adaptability and flexibility for modern communication devices. In this work, we present a conceptual model for a single-antenna N-port (SANP) FAS over spatially correlated Nakagami-m fading channels and compare it with the traditional diversity schemes in terms of outage probability.
The proposed FAS model switches to the best antenna port and resembles the operation of a selection combining (SC) diversity.
FAS improves the quality of service (QoS) of the network through antenna port selection. The advantage of FAS is the ability to fit hundreds of antenna ports into a half-wavelength antenna size at the cost of spatial channel correlation. Simulation results demon- strate the superior outage probability performance of FAS at several tens of antenna ports compared to the traditional diversity schemes such as maximum ratio combining, equal gain combining, and SC. Moreover, the novel probability and cumulative density functions for the land mobile correlated Nakagami-m random variates are evaluated in this paper.
Index Terms—Fluid antenna system (FAS), Nakagami-mfading, outage probability, single-antennaN-port (SANP) FAS.
I. INTRODUCTION
The high quality of service (QoS) is one of the metrics in modern communication systems that motivate for new techno- logical products and maintain revenue for telecom providers.
Future communication systems require innovative fabrication solutions on the radio frequency (RF) chain components, specif- ically on antennas, to upgrade the performance of multiple- input multiple-output (MIMO) systems. The main limitation of MIMO antennas is the small physical size of devices to accommodate a half-wavelength separation requirement be- tween antenna elements. For instance, IoT devices can only accommodate up to 16 antennas due to the limited physical space in millimeter wave (mmWave) frequency bands [1].
As a remedy, mechanically flexible fluid antennas have been offered for 6G communication, where a number of antenna ports are no longer an issue for mobile and IoT devices. Fluid antennas can be built on fluid materials (e.g., Mercury, Eutectic Gallium-Indium, Galinstan, etc.) and controlled by software [2]. Moreover, tunable fluid antennas may operate over a wide range of frequencies, enabling cognitive communication for IoT and mobile devices. A review on fluid antennas, including their materials, fabrication techniques, and applications, have
been studied in [3]. According to [3], fluid antennas are free of mechanical deformations such as bending and stretching, which is a main physical limitation of conventional copper antennas. Materials in fluidic antennas are classified into non- conductive (acetone, ethyl acetate, etc.), partially conductive (seawater), and conductive. Moreover, fluid antennas are built on soft substrates to maintain elastic antenna properties. More- over, the authors in [4] discussed the state-of-the-art, design configuration, advantages, and limitations of the fluid antennas.
The authors in [5] evaluated the approximate outage proba- bility performance for single-antennaN-port (SANP) FAS over spatially correlated Rayleigh fading channels and compared its performance to theL-antenna maximal ratio-combining (MRC) receiver. According to this study, SANP FAS outperforms the L-antenna MRC receiver given thatN is a large number. More- over, theL-diversity RF chain cost and power supply could be reduced by using the FAS. The same authors studied ergodic capacity and lower bound for ergodic capacity performance as well as level crossing rate and average fade duration for the same physical FAS model in [6]. Based on this study, the FAS model was built on a small space and achieved the capacity performance of the MRC antenna system. The authors also showcased a possible architecture for the N-port FAS.
Furthermore, a working prototype of a circular patch liquid microstrip antenna that uses mercury as a radiating element was reported in [7]. In addition, a3D-printed fluidic monopole antenna system that resonates between 3.2 to 5 GHz was designed in [8]. The authors in [9] presented another working prototype for a frequency reconfigurable antenna architecture utilizing dielectric fluids.
Motivated by [5] and [6], we study the SANP equally- spaced antenna system that is exposed to correlated Nakagami- mfading channels. All antenna ports are correlated with respect to the first port by using the land mobile correlation model as in [2], [6]. One of the popular correlation models is the exponential one, which accounts for the correlation of adjacent antenna ports, as was discussed in [10]. However, this model is more applicable when multiple antenna ports are active simultaneously and correlation occurs between two adjacent antenna ports. The previous works [2], [5] and [6] studied
Rayleigh fading channel for the SANP FAS model. Although Rayleigh fading statistics simplify the analysis, it does not model line-of-sight (LOS) components between communicating nodes and only considers small-scale scattering that better suits to model the non-LOS (NLOS) signal component. According to the authors in [11] and [12], Nakagami-mfading along with analog beamforming well models the mmWave channel.
In this paper, we introduce the FAS as a method to enhance the system QoS in terms of outage probability. The key contri- butions of this work are listed as follows.
• We derive the joint probability density function (PDF) and cumulative density function (CDF) of the received signal at the FAS model given spatially correlated Nakagami- m fading channels. Based on it, we further evaluate the outage probability formula in a single integral form. These are novel analytical derivations as no prior work studied correlated Nakagami-mchannel for the FAS model.
• Numerical analyses on the Nakagami-mfading parameter, antenna length, and signal-to-noise-ratio (SNR) thresh- old are investigated in terms of outage probability. It is noted that the Nakagami-m fading parameter is a major contributing factor to the outage probability performance.
Furthermore, the SNR threshold has a moderate effect on the outage probability, whereas antenna length has the least contribution to the system performance.
• We also compare the outage probability performance of FAS with the conventional diversity techniques such as MRC, equal gain combining (EGC), and selection combining (SC). The FAS model demonstrated a higher performance while using several tens of antenna ports.
FAS enhances the outage probability performance of the wireless networks, thus, increases QoS experienced by users.
II. SYSTEMMODEL
In this work, we study a system model that comprises a single-antenna transmitter (Tx) and SANP FAS-enabled re- ceiver (Rx) built on a fluid material and connected to a single RF chain as shown in a hypothetical device in Fig. 1. A liquid metal is encapsulated into the microfluidic channel of the length d, and an antenna at location portkis treated as an ideal point antenna. We assume that FAS is able to switch to the strongest antenna port among availableN ports evenly distributed over d = αλ, whereλis a wavelength of the operating frequency, andαis a positive coefficient. Since antenna ports are located at close proximity to each other, the correlation phenomenon is an inevitable factor for the FAS model. Inspired by [5], we assume that (N −1) ports are correlated to its first port by using the land mobile correlation model [13, Table 2.1].
We consider Nakagami-m fading for the proposed system model. The Nakagami-m distribution is a widely used model [14] that describes physical fading radio channels through the m parameter that determines the severity and existence of the direct communication link. Moreover, Nakagami-mdistribution shows a good fit with the empirical data [15]. The authors in
Fig. 1:A system model of SANP FAS with a antenna length,d.
[16] suggested parameterization of the correlated Nakagami- m random variables (RVs) by using the square root of a summation of m sqaured Rayleigh fading RVs to generate an envelop of the correlated Nakagami-m RV. Since N complex Gaussian RVs with zero mean and1/2variance are required to implement the selection amongN-port Rayleigh RVs, it will be necessary to apply m×N complex Gaussian RVs to model a correlation in Nakagami-mchannels in the SANP FAS model.
The channel at portk|∀k∈A, withA={1, . . . , N}, is modeled ashk =Pm
l=1|Hkl|2. Thus, the spatially correlated Nakagami- mfading channel at port kis parameterized as
Hkl = q
(1−µ2k)xkl+µkx0l +j
q
(1−µ2k)ykl+µky0l
, l={1, . . . , m}, (1) where xkl and ykl are independent Gaussian RVs with zero mean and 1/2 variance. Moreover, the correlation coefficient, denoted by 0< µk <1, can be found as [13]
µk=J0
2π(k−1)α N−1
, for k={2, . . . , N}, (2) whereJ0(·)is the zero-order Bessel function of the first kind.
The received signal at the antenna port k is formulated as yk =hks+w, wherehk is a Nakagami-mchannel amplitude, sis a transmitted signal, and wis an additive white Gaussian noise, with zero mean and σw2 variance. Note that a unity distance is assumed between Tx and Rx to focus on the main results. The PDF of the Nakagami-mRV is given by [17]
frk(rk) = 2mmrk2m−1e−
mr2 k σ2
k
Γ(m)σk2m , (3) where Γ(·) is the Gamma function, m ≥ 0.5 represents the severity fading parameter, andσk2is the average channel power.
III. OUTAGEPROBABILITY
By definition, the outage probability measures the probability that the received SNR that falls below a predefined threshold.
As mentioned before, FAS can switch to one of the strongest antenna port channel located along d = αλ length. The strongest antenna channel is denoted as max{|h1|,|h2|,· · ·,|hN|}, where |h1| is taken as a reference
port and the other (N −1) ports are correlated with respect to it. Hence, outage probability for SANP FAS over spatially correlated Nakagami-mfading channel is generalized as
Pout(γth)
= Pr
|h1|2<γth
γ ,|h2|2< γth
γ · · ·,|hN|2<γth γ
, (4) where γth is the SNR-associated threshold, γ is the average received SNR estimated by γ = E{|hk|2σ}2E{|s|2}
w , and E{·}
stands for the expectation operator.
To evaluate the outage probability, we need to derive the PDF and CDF of joint Nakagami-mRVs, which are calculated in Proposition 2 and Theorem 1, respectively. It is essential to obtain the PDF of two joint Nakagami-m variates that are spatially correlated with respect to the port 1 to find the general PDF formula for N correlated Nakagami-m variates.
Proposition 1 evaluates the conditional PDF of two jointly distributed Nakagami-mRVs.
Proposition 1: Let|h1|and|h2|be non-negative Nakagami- mRVs. A conditional probability of|h2|on|h1|is given by
f|h2|
|h1|(r2|r1) =2mr1−m1 rm2σm−11 e−
m 1−µ2
r2 2 σ2
2e−
µ2 2m 1−µ2
r2 1 σ2 1
σm+12 (1−µ22)µm−12
×Im−1
2mµ2r1r2
(1−µ22)σ1σ2
, (5)
whereIm−1(·)is the(m−1)th order modified Bessel function of the first kind, µ2is the correlation coefficient of the second port, andσ2k|k∈{1,2} is the average signal power.
Proof: The proof is presented in Appendix A.
By making use of Proposition 1, the joint PDF of N correlated Nakagami-mRVs is evaluated as below.
Proposition 2: The joint PDF of N spatially correlated
|hk|k∈A| Nakagami-mRVs is computed as f|h1|,···,|hN|(r1,· · · , rN) =2r12m−1mm
Γ(m)σ12m e−
r2 1m σ2
1
×
N
Y
k=2
2mr11−mrmk σ1m−1 σkm+1(1−µ2k)µm−1k e−
mσ2 1r2
k+mσ2 kr2
1µ2 k σ2
1σ2 k(1−µ2
k)
×Im−1
2mµkr1rk
σ1σk(1−µ2k)
. (6)
Proof: The proof is given in Appendix B.
The next corollary estimates the joint PDF for theN uncor- related Nakagami-mfading channels.
Corollary 1:Now, we will consider a case with no correlation between channels, i.e.,µk= 0. However, a simple substitution ofµk = 0into (6) will lead to an undefined result because of the
“0/0” term. Therefore, we proceed by expanding the modified Bessel function term in terms of its series representation as in [18, (8.445)] and get the following identity for limiting behavior
µlimk→0
1 µm−1k Im−1
2mµkr1rk
σ1σk(1−µ2k)
= 1 Γ(m)
mr1rk
σ1σk
m−1
. (7)
By utilizing (7) and taking the limit in (6), one can obtain f|h1|,···,|hN|(r1, r2,· · · , rN) =
N
Y
k=1
2mmrm−1k Γ(m)σk2me−
r2 km σ2
k . (8) This means that if there is no correlation; thus, the combined PDF equals the product of individual PDFs obtained using (3).
The CDF of N jointly distributed Nakagami-m RVs is assessed in the next theorem.
Theorem 1: The joint CDF of N correlated Nakagami-m RVs is calculated as
F|h1|,|h2|,···,|hN|(r1, r2,· · · , rN) = 2mm Γ(m)σ12m
R1
Z
0
r2m−11 e−
mr2 1 σ2
1
×
N
Y
k=2
"
1−Qm
s 2mµ2kr21 σ21(1−µ2k),
s 2mR2k σk2(1−µ2k)
!#
dr1, (9) whereQmis the Marcum Q-function.
Proof:The proof is relegated in Appendix C.
The following key theorem delivers the outage probability expression for the system under study.
Theorem 2: Suppose |hk|k∈A| are correlated Nakagami-m RVs. The outage probability for SANP FAS is evaluated as
Pout(γth) = 2mm Γ(m)σ12m
qγ
th γ
Z
0
r2m−11 e−
mr2 1 σ2
1
×
N
Y
k=2
1−Qm
s 2mµ2kr21 σ12(1−µ2k),
s 2mγth σk2(1−µ2k)γ
!!
dr1. (10) Proof: By setting R1 =R2 =· · ·=RN =qγ
th
γ into the CDF given in (9), we get the outage probability formula.
IV. NUMERICALRESULTS ANDDISCUSSION
This section is devoted to study the outage probability performance of the proposed system model for different fading, threshold, and antenna length parameters. As a rule of thumb, an improvement of outage probability directly enhances the network QoS. In Fig. 2, the performance of 100-port FAS is studied for the fading parametersm={1,2,4,6} and antenna sizes α={1,5} at the thresholdγth = 5dB. First, it is ob- served that a five-times increase in antenna size has resulted in the improved outage performance. For instance, when m= 4, N = 10, a curve with α = 1 reaches Pout = 1.2×10−4, whereas a curve with α = 5 attains Pout = 2.7 ×10−5. Secondly, this figure demonstrates a crucial impact of fading parameters on the outage probability. The NLOS and LOS cases are modeled by setting m= 1 and m ={4,6}, respectively.
To attain Pout = 0.05 withm ={1,2,4,6} channel models, FAS requiresN ={90,18,3,2}ports, correspondingly. Hence, FAS atm={4,6} would require30- and 45-times less ports compared to the NLOS one.
1 10 100 10
-5 10
-4 10
-3 10
-2 10
-1 10
0
Outageprobability
Number of ports = 1
= 5
m = 4
m = 2 m = 1
m = 6
Fig. 2:Outage probability, whenm={1,2,4,6}, α={1,5}, and γth= 5dB.
1 10 100
10 -5 10
-4 10
-3 10
-2 10
-1 10
0
th
= {0, 1, 2, 3} dB
Outageprobability
Number of ports m = 4
m = 1 th
= {0, 1, 2, 3} dB
Fig. 3:Outage probability, whenγth={0,1,2,3}dB,α= 0.5, and m={1,4}.
In Fig. 3, we study how the SNR threshold values γth = {0,1,2,3} dB affect the outage probability performance at the LOS/NLOS channel cases while FAS size is α = 0.5.
As expected, lower SNR threshold values produce lower out- age probabilities and thus improve the system performance for both LOS/NLOS cases. For example, at m = 1 and N = 30, we record Pout = {9×10−5,0.001,0.01,0.05}
for the corresponding γth = {0,1,2,3} dB. For each dB unit increase, there is, at least, five-times outage probability performance degradation. Moreover, we observe a significant outage performance difference between the LOS and NLOS cases at the same SNR thresholds. For instance, at γth = 1dB and N = 3, the m= 4curve results inPout= 10−4, and the m= 1curve shows Pout= 0.4.
In Figs. 4-5, we analyze how FAS is compatible with the conventional MRC [19, (3)], EGC [20, (3)], and SC [21]
diversity techniques under the LOS/NLOS channel conditions.
More precisely, in Fig. 4, we study FAS with the sizeα= 0.5
1 10 100
10 -2 10
-1 10
0
Outageprobability
Number of ports MRC, m = 1
FAS, m = 1,
SC, m = 1
EGC, m = 1 L = 4 L = 3 L = 2
L = {2, 3, 4, 5}
L = 5
Fig. 4:Outage probability, whenγth= 53dB,α= 0.5,m= 1, and MRC/EGC/SC receivers withL={2,3,4,5}.
1 10 100
10 -6 10
-5 10
-4 10
-3 10
-2 10
-1
Outageprobability
Number of ports MRC, m = 4
EGC, m = 4
SC, m = 4
FAS, m = 4,
L = 2
L = 3
Fig. 5:Outage probability, whenγth= 5dB,α= 0.5,m= 4, and MRC/EGC/SC receivers withL={2,3,4,5}.
atm= 1 andγth = 4 dB. The dashed, solid and dash-dotted lines represent the MRC, EGC and SC receivers, respectively, with L={2,3,4,5} branches that define the diversity order.
This figure illustrates how many ports are necessary for the FAS scenario to outperform the conventional diversity methods.
Thus, FAS outperforms the order-2 MRC and EGC diversity schemes, when the system is deployed withN≥5andN ≥3 ports, respectively. Likewise, FAS surpasses the L = 5-order MRC and EGC curves, when the number of ports is above N = 40 and N = 28, accordingly. A complete comparison table between FAS and the conventional diversity receivers is presented in Table I.
In Fig. 5, we analyze the communication at the LOS scenario, whenm= 4,γth= 4, andα= 0.5. From this figure, FAS with four ports excels the L = 2- order MRC and EGC schemes.
Likewise, FAS outperforms the SC scheme, when the system is equipped with, at least, two ports. Similarly, FAS beats the L = 3-order MRC, EGC, SC scenarios, when N > 11,
Table I: The amount of FAS ports required to outperform the MRC/EGC/SC techniques.
Order of diversity Qty per MRC/EGC/SC
L = 2 5/3/2
L = 3 11/8/3
L = 4 24/17/5
L = 5 40/28/6
N >10, and N >2, respectively. It is pertinent to note that FAS (equipped with dozens of ports evenly distributed over half-wavelength antenna space) is able to achieve a better out- age performance than its three counterparts at the LOS/NLOS scenarios. Moreover, note that the MRC and EGC diversity schemes require channel state information and L-order RF- chain components compared to FAS, which needs only a single RF chain as the SC scheme does. Every additional diversity order for MRC and EGC requires an extra cost of RF chains.
On the other hand, SC demonstrates the worst performance compared to the other diversity schemes and FAS.
Fig. 6 illustrates the outage performance for the MRC and FAS scenarios, with different transmit SNR values at the threshold γth = 10 dB and m = 2. We consider MRC, withL ={2,3,4}, and FAS, with N = {100,200,300,500}
antenna ports spread over the space defined byα={1,1.5}.
We assume that MRC is characterized by a λ/2 antenna separation maintained between the two neighbouring antennas.
Therefore, we have selected the proportional FAS antenna sizes of α = {1,1.5} that directly correspond to MRC with L = {3,4}. From this plot, N = 500-port FAS outperforms the L = 2 MRC above 4.12 dB and theL = 3 MRC above 5.14dB of transmit SNR, correspondingly. However, the FAS performance is worse than the one of MRC with L = 4.
There is a notable outage metric improvement when the antenna port size rises from 100 to 500 antenna ports. For instance, at Pout = 0.001, FAS with α = 1.5 on average requires at least0.54dB SNR for every100 ports. Next, the antenna size has demonstrated a minor influence on the outage probability performance as opposed to the number of antenna ports.
V. CONCLUSION
Traditional conductive antennas often are not suitable for cer- tain applications that requires flexibility and reconfigurability of electromagnetic performance. FAS offers many advantages as low cost, small size, flexibility, and high performance. In this work, we derived the outage probability performance of SANP FAS over spatially correlated Nakagami-m fading channels.
Previous works have not analyzed correlated Nakagami-m fading channels for the proposed system model. The FAS model with the order of tens ports demonstrated a higher performance than the MRC and EGC diversity schemes of theL ={2,3}
order. Moreover, FAS requires less than ten ports to achieve the performance of the SC scheme. Thus, it is possible to increase the network QoS by increasing the number of antenna ports.
The future work aims to build a hardware prototype of FAS to justify its advantage over traditional antenna diversity schemes.
0 2 4 6 8 10
10 -4 10
-3 10
-2 10
-1 10
0
N = 100 N = 200 N = 300
Outageprobability
Transmit SNR, dB
FAS
FAS 1.5
MRC
L = {2, 3, 4}
N = 500
Fig. 6: Outage probability, whenγth= 10 dB,α={1,1.5}, N = {100,200,300,500},m= 2, and MRC receivers withL={2,3,4}.
APPENDIXA PROOF OFPROPOSITION1.
The PDF f
|h2|
|h1|(r2|r1) of the two-port FAS conditioned on port1is computed by taking a ratio of the joint probability of two RVsf|h1|,|h2|(r1, r2)over the PDF of|h1| as
f|h2|
|h1|(r2|r1) = f|h1|,|h2|(r1, r2)
f|h1|(r1) , (A.1) where the joint PDF of two correlated Nakagami-mvariates is given in [13, (6.1)] as
f|h1|,|h2|(r1, r2) =4mm+1(r1r2)me−
m 1−µ2
2
r2 1 σ2 1
+r
22 σ2 2
Γ(m)σ12σ22(1−µ22)(σ1σ2µ2)m−1
×Im−1
2mµ2r1r2
(1−µ22)σ1σ2
. (A.2)
By taking the ratio of (A.2) and (3), the conditional probability for spatially correlated two ports is calculated in (5).
APPENDIXB PROOF OFPROPOSITION2.
First, the PDF of multiple Nakagami ports conditioned on port1 is the product of terms in (5)
f|h2|,···,|hM|
|h1|(r2,· · · , rM|r1) =
N
Y
k=2
f|hk|
|h1|(rk|r1)
=
N
Y
k=2
2mr1−m1 rmkσm−11 e−
m 1−µk
r2 k σ2 ke
m 1−µ2
k r2
k σ2 k
σ2k(1−µ2k)(σ1σkµk)m−1
×Im−1
2mµkr1rk
(1−µ2k)σ1σk
. (B.1)
Next, by multiplying (3) and (B.1), we obtain the joint PDF for N Nakagami-mvariates in (6).
APPENDIXC PROOF OFTHEOREM1.
To find the CDF, the following integral needs to be evaluated F¯ =F|h1|,|h2|,...,|hN|(r1, r2, . . . , rN)
= Z R1
0
· · · Z RN
0
f|h1|,···,|hN|(r1, . . . , rN)dr1· · ·drN. (C.1) By using [18, (8.445)], substituting (6) into (C.1), and perform- ing some algebraic simplifications, we get
F¯= 1 Γ(m)
N
Y
µk=11=0
2mm (1−µ2k)mσ2mk
× Z R1
0
r2m−11 e−
m σ2 1
+PN k=2
mµ2 k σ2
1(1−µ2 k)
r12
×
" N Y
k=2
∞
X
ik=0
1 ik!Γ(m+ik)
mµkr1 σ1σk(1−µ2k)
2ik
× Z Rk
0
r2m−1+2ik ke−
m σ2
k(1−µ2 k)r2k
drk
#
dr1. (C.2) Using [18, (3.381.1)] and applying the change of variables, the internal integral in terms ofrk can be evaluated as
Z Rk 0
r2m−1+2ik ke−
m σ2
k(1−µ2 k)r2k
drk
= 1 2
m σ2k(1−µ2k)
−m−ik
γ
m+ik, mR2k σk2(1−µ2k)
, (C.3) whereγ(·,·)is the lower incomplete gamma function. If (C.3) is substituted into (C.2) and then some algebraic manipulations are performed, we get
F¯ = 2mm Γ(m)σ2m1
Z R1 0
r2m−11 e−
m σ2 1
+PN k=2
mµ2 k σ2
1(1−µ2 k)
r21
×
" N Y
k=2
∞
X
ik=0
r2i1k ik!Γ(m+ik)
mµ2k σ1(1−µ2k)
ik
×γ(m+ik, mR2k σk2(1−µ2k))
dr1. (C.4)
We note that the integral in (C.4) can not be evaluated due to the presense of a sum inside the product, which also contains the integration variable r1. By using [18, (8.356.3)] and the definition of the Marqum Q-function
Qm(α, β) =e−α
2 2
∞
X
ik=0
1 ik!
α2 2
ik
Γ(m+ik, β2/2) Γ(m+ik) , (C.5) where in our case α2 = σ2mµ2 2kr21
1(1−µ2k) and β2 = σ22mR2k k(1−µ2k), we rewrite (C.4) as the final CDF expression shown in (9).
Remark: The validity of (9) or (C.4) can be checked as follows. Let us consider two extreme cases.
Case 1: we assume that all integration limits are defined as R1 =. . .=RN =∞. Their substitution into (C.4) precisely
results inF¯ = 1.
Case 2: we assume that all integration limits are zero, i.e., R1=. . .=RN = 0; this leadsF¯ = 0. Therefore, both outputs support the domain of the outage metric.
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